Open queueing network model of a computer system: completed jobs

Transkript

1 E Queueing Networks E Queueing Networks Open queueing network model of a computer system: Stream of new jobs Disk CPU Printer Stream of completed jobs Magnetic Tape E.164 E Queueing Networks "Central-Server-Model (CSM)" : Computer system modelled as a closed queueing network µ 2 µ 1 µ 3 µ N E.165

2 E.1 Description E.1 Description Notation N Number of nodes K The constant number of jobs in a closed network (k 1, k 2,...,k N ) The state of the network k i The number of jobs at the ith node; for closed networks k i = K i=1 m i The number of parallel servers at the ith node (m i 1) µ i Service rate of the jobs at the ith node 1/µ i The mean service time of the jobs at the ith node E.166 E.1 Description p ij p 0j p i0 λ 0i λ λ i Routing probability, the probability that a job is transferred to the jth node after service completion at the ith node. (In open networks, the node with index 0 represents the external world to the network.) The probability that a job entering the network from outside first enters the jth node The probability that a job leaves the network just after completing service at node i (p i0 =1 N p ij ). The arrival rate of jobs from outside to the ith node The overall arrival rate from outside to an open network (λ = N λ 0i ) i=1 the overall arrival rate of jobs at the ith node E.167

3 E.1 Description Fundamental formulas: Traffic equations (open networks): λ i = λ 0i + λ j p ji, for i =1,...,N Traffic equations (closed networks): λ i = λ j p ji, for i =1,...,N E.168 Visit ratio (mean number of visits, relative arrival rate): E.1 Description λ: Overall throughput of the network Open networks: e i = λ i λ, for i =1,...,N e i = p 0i + e j p ji, for i =1,...,N Closed networks: e i = e j p ji, for i =1,...,N E.169

4 E.2 Performance Measures E.2 Performance Measures Steady-state probabilies: π(k 1,...,k N ) Probability, network is in state (k 1,...,k N ) Marginal probability ( P(k jobs in node i) ): Open networks: π i (k) = π(k 1,...,k N ) k i =k Normalizing condition: π(k1,...,k N )=1 E.170 E.2 Performance Measures Closed networks (number of jobs is constant): k j = K with (0 k j K) π i (k) = π(k 1,...,k N ) NP k j =K & k i =k Normalizing condition: π(k 1,...,k N )=1 NP k j =K E.171

5 E.2 Performance Measures Utilization of node i: m i = 1: ρ i = π i (k) k=1 or: ρ i =1 π i (0) m i > 1: ρ i = 1 min (m i,k)π i (k) =1 m i or: k=0 m i 1 k=0 m i k π i (k) m i ρ i = λ i m i µ i E.172 E.2 Performance Measures Throughput of node i: λ i = m i ρ i µ i Load dependent node (Service rate depends on the number of jobs in the node): λ i = π i (k)µ i (k) k=1 Example for a load dependent node M/M/m-node: µ i (k) =min(k, m i ) µ i µ i : Service rate of a single server E.173

6 E.2 Performance Measures Overall throughput: Open network: λ = i=1 λ 0i Closed network: λ = λ i e i E.174 E.2 Performance Measures Mean number of jobs in node i: K i = Little s law: k π i (k) k=1 K i = λ i T i Mean queue length of node i: Q i = (k m i ) π i (k) k=m i Little: Q i = λ i W i E.175

7 E.2 Performance Measures Mean response time of node i (Little): T i = K i λ i Mean overall response time (Little): T = K λ Mean waiting time of node i: W i = T i 1 µ i E.176 Multiple job classes in the network Jobs of different classes have different service rates and different routing probabilities (transition probabilities) Networks with open and closed classes are called mixed networks: Source Open Class Closed Class µ 1 µ 2 Sink E.177

8 1 Notation R k ir The number of job classes in a network Thenumberofjobsoftherth class at the ith node; for a closed network: R k ir = K (0.1) i=1 r=1 K r Thenumberofjobsoftherth class in the network; not necessarily constant, even in a closed network: k ir = K r (0.2) i=1 K The number of jobs in the various classes, known as the population vector (K =(K 1,...,K R )) E.178 S i The state of the ith node (S i =(k i1,...,k ir )): S i =K (0.1) i=1 S The overall state of the network with multiple classes (S =(S 1,...,S N )) µ ir The service rate of the ith node for jobs of the rth class p ir,js The probability that a job of the rth class at the ith node is transferred to the sth class and the jth node (routing probability) p 0,js p ir,0 The probability in an open network that a job from outside the network enters the jth node as a job of the sth class The probability in an open network that a job of the rth class leaves the network after having been serviced at the ith node, so: p ir,0 =1 s=1 R p ir,js (0.2) E.179

9 λ The overall arrival rate from outside to an open network λ 0,ir The arrival rate from outside to node i for class r jobs (λ 0,ir = λ p 0,ir ) Thearrivalrateofjobsoftherth class at the ith node: λ ir λ ir = λ p 0,ir + s=1 R λ js p js,ir ; (0.1) for closed networks, p 0,ir =0(1<i<N,1 <r<r) and we obtain: R λ ir = λ js p js,ir. (0.2) s=1 E.180 The mean number of visits e ir of a job of the rth class at the ith node of an open network can be determined from the routing probabilities: R e ir = p 0,ir + e js p js,ir, s=1 for i =1,...,N, r =1,...,R. (0.1) For closed networks, the corresponding equation is: R e ir = e js p js,ir, s=1 for i =1,...,N, r =1,...,R. (0.2) Usually we assume that e 1r =1,forr =1,...,R, although other settings are also possible. E.181

10 2 Performance Measures Steady-state probability: π(s 1,...,S N ) Probability, network is in state (S 1,...,S N ) Marginal probability ( P(node i is in state k) = P(S i = k) ): Open networks: π i (k) = s i =k π(s 1,...,S N ) Closed networks: π i (k) = NP S j =K &S i =k π(s 1,...,S N ), E.182 Utilization of node i with jobs of class r: ρ ir = 1 m i all states k with k r > 0 π i (k) k ir k i min(m i,k i ), k i = R r=1 k ir or: λ ir ρ ir = m i µ ir E.183

11 Throughput of node i with jobs of class r: λ ir = all states k with k r > 0 π i (k) k ir k i µ i (k i ) or: λ ir = m i ρ ir µ ir Overall throughput of the network with jobs of class r: Open network: Closed network: λ r = i=1 λ 0,ir λ r = λ ir e ir E.184 Mean number of jobs of class r in node i: Little: K ir = all states k with k r > 0 k r π i (k) K ir = λ ir T ir Mean queue length of jobs of class r in node i: Q ir = λ ir W ir E.185

12 Mean response time of jobs of class r in node i: T ir = K ir λ ir Mean waiting time of jobs of class r: W ir = T ir 1 µ ir E.186